State-Constrained Continuous-Control Zero-Sum Dynamic Games with One-Sided Payoff Information

We study $G$, a $K$-stage zero-sum dynamic game with continuous and simultaneous controls, one-sided payoff information, and public state constraints. $G$ arises in defense, sports, finance, among other agentic competitions. Nash equilibrium strategies in $G$ may mix over a continuum of controls, producing an intractable game tree. To this end, we introduce a pair of information-relaxing Stackelberg variants of $G$ where one player best responses to their opponent's next control, and prove that Nash leaders in these variants use at most $I$ control atoms at any information states. Hence the game tree complexity of these variants are at most $I^K$. We then show that under sufficient game-theoretic and geometric assumptions, this pair of atomic leader strategies have vanishing exploitability in $G$ as the time grid refines. This result justifies a computationally tractable approximation of the Nash of $G$ via atomic strategy restriction and bounded control-atom insertion. Effectiveness of our method is demonstrated through an analytical Hexner's game and a car chasing game with nonlinear dynamics and nonconvex state constraints.